suppose we are told that a number has the following remainde

suppose we are told that a number has the following remainder

0 mod 10

2 mod 11

7 mod 19

we are also told that the number is greater than 0 and less than the product 10*19*11=2090. What\'s that number

let the numbe be 10x

given the number has remainders 0,2,7 when divided by 10,11,19

given when divided 11 remainder is 2

10x=11y+2

10x=19z+7

11y+2=19z+7

11y-19z=5

divide both sides with 11 we get

y-19/11z=5/11

y-z-8/11z=5/11

we know that number are integers so fractional part should be equal to fractional part

8/11z=5/11

8/11z-5/11=k(assume let it be k)

z=(11k+5)/8

and we other condition that number should be multiple of 10

z=10l(assume)=(11k+5)/8

k=(80l-5)/11 now substitute the valves of l such that k is integer l=0,1,2,3,,4,5,6,7,8,9.....

for l=9 k becomes integer

k=80*9-5/11=720-5/11=715/11=65

substitute valve of k in z we get z=11*65+5/8=90

the number =19z=19*90=1710